GIFT  OF 


t 


SUGGESTIONS 

ON  THE 

TEACHING   of  GEOMETRY 

With  Especial  Reference  to  the  Use  of 
DURELL  and  ARNOLD'S  GEOMETRY 


BY 

FLETCHER   DURELL,  Ph.D. 

HEAD  OP  MATHEMATICAL  DEPARTMENT,  THE  LAWRENCE- 

VILLE  SCHOOL;  AUTHOR  OF  THE  DURELL  MATHEMATICS 

SERIES  AND  JOINT  AUTHOR  OF  THE  DURELL- 

ARNOLD  MATHEMATICS  SERIES 


CHARLES  E.  MERRILL  COMPANY 
NEW  YORK  CHICAGO 


CONTENTS 


PAGE 

1.  INTRODUCTORY  REMARKS 1 

2.  TEACHING  OF  DEFINITIONS,  ELEMENTARY  PRINCI- 

PLES    1 

3.  TYPICAL  RECITATION  ON  THEOREMS 3 

4.  VARIATIONS  IN  TYPICAL  RECITATION  ON  PROPOSI- 

TIONS    5 

5.  METHODS  OF  USING  THE  FINAL  FIFTEEN  MINUTES.  6 

6.  FIRST  THEOREMS 9 

7.  FIRST  ORIGINALS 10 

8.  TYPICAL  RECITATION  ON  ORIGINALS 12 

9.  PRACTICAL  APPLICATIONS  OF  GEOMETRY 14 

10.  ORNAMENTAL  DESIGNS 17 

11.  NUMERICAL  EXERCISES 18 

12.  USE  OF  ALGEBRA  IN  GEOMETRIC  STUDY 19 

13.  DEFICIENCY  STUDY 21 

14.  STUDY  OF  SOLID  GEOMETRY ,  22 


46881.4 


SUGGESTIONS  ON  THE  TEACHING  OF 
GEOMETRY  * 

WITH  ESPECIAL  REFERENCE  TO  THE  USE  OF 

DURELL  AND  ARNOLD'S  GEOMETRY 

1.  Introductory  Remarks. — In  recent  years  pupils  in 
the  high  school  on  the  average  are  noticeably  more  immature 
than  they  were  ten  years  ago.  The  subject  matter  also  of 
geometry  has  been  changed,  and  to  a  certain  extent  voca- 
tionalized  and  humanized.  These  changes  call  for  modi- 
fications in  class-room  methods  of  teaching  geometry,  in 
order  to  obtain  maximum  results.  It  is  hoped  that  the 
suggestions  which  follow  may  help  to  meet  the  new  situation. 

It  is  to  be  understood  that  this  pamphlet  is  written  merely  by  way 
of  suggestion  and  for  teachers  who  desire  any  help  they  can  get  from 
any  source.  It  is  not  intended  for  teachers  who  have  already  worked 
out  methods  of  their  own  adequate  to  their  needs.  Suggestions  of 
any  kind  looking  to  the  betterment  of  the  following  remarks  will  be 
gladly  received. 

While  the  mind  of  the  average  high  school  pupil  is 
immature  and  occupied  with  many  interesting  concrete 
things  such  as  the  movie,  kodak,  and  automobile,  it  is  also 
alert,  eager,  and  quick  to  grow  when  interested.  Hence 
the  best  general  course  to  follow  in  teaching  such  pupils  is 
to  give  them  at  the  outset,  a  large  amount  of  simple  and 
easily  appreciated  work.  This  arouses  natural  growth 
processes  in  their  minds,  so  that  in  time,  and  often  without 
serious  effort,  they  develop  the  power  to  do  more  difficult 
work  and  form  an  active  appetite  for  it. 

2.  The  Teaching  of  Definitions  and  Elementary  Prin- 
ciples.— Under  present  conditions,  in  teaching  the  first 
*  Copyright,  1921,  by  Charles  E.  Merrill  Co, 


2  ,  '  TEACHING  OF  GEOMETRY 

groups  of  definitions  in  geometry,  it  is  usually  best  at  first 
to  concentrate  attention  on  certain  leading  or  specially 
important  ones,  as  on  the  definitions  of  geometry,  curved 
line,  parallel  lines,  circle,  right  angle,  perpendicular,  and 
complementary  and  supplementary  angles,  on  pp.  7-20  of 
Durell  and  Arnold's  Plane  Geometry.  The  other  defini- 
tions and  discussions  on  these  pages  are  to  be  read,  con- 
considered,  queried,  and  commented  upon,  till  they  are  fully 
understood,  but  attention  is  to  be  concentrated  on  the  list 
given  above,  the  ideal  result  aimed  at  being  that  each  of 
these  pivotal  definitions  shall  be  so  thoroughly  memorized 
that  each  member  of  the  class  shall  be  able  to  give  it  instantly 
at  any  time  and  shall  not  be  able  to  forget  it.  When  these 
definitions  have  been  thoroughly  assimilated,  they  form  a 
foundation  or  starting  point  to  which  the  neighboring 
definitions  can  be  readily  attached.  Some  of  these  will 
come  to  the  pupils  as  natural  consequences  of  those  first 
learned  and  can  readily  be  picked  up  in  review. 

The  learning  of  definitions  like  that  of  vertical  angles 
may  be  postponed  till  Prop.  I  is  studied/  In  like  manner 
the  geometric  axioms  and  postulates  (p.  23)  in  the  first 
study  are  to  be  read  and  discussed  but  not  committed  to 
memory  till  later,  as  they  are  used  in  the  course  of  demonstra- 
tions. - 

Similarly  the  principles  enunciated  on  pp.  26-28,  as 
matters  of  direct  inference,  are  not  to  be  committed  to 
memory  in  a  mass,  but  each  is  to  be  learned  in  connection 
with  its  use. 

Thus,  in  working  the  examples  in  Group  8,  each  principle  as  it  is 
taken  from  §§55-68  and  used,  is  to  be  quoted  in  exact  language. 

It  is  usually  best  to  hav^  pupils  learn  corollaries  not  as 
they  occur  as  consequences  of  propositions,  but  as  they  are 
used  in  connection  with  later  propositions. 


TEACHING  OP  GEOMETRY  3 

This  is  not  only  a  great  economy  of  time,  but  is  also  an 
illustration  -of  the  guiding  principle  of  the  book;  viz., 
that  geometry  is  not  merely  a  set  of  correct  deductions, 
but  is  a  series  of  improving  tools,  the  relation  of  each  of 
which  to  its  uses  should  be  made  as  close  as  possible. 

By  following  the  above  method,  pp.  7-28  of  the  textbook  can  be 
covered  in  five  lessons.  In  the  last  ten  or  fifteen  minutes  of  each  reci- 
tation, have  the  class  do  written  work,  such  as  defining  certain  terms, 
or  making  simple  drawings,  like  those  on  page  14,  or  solving  examples 
like  those  in  Groups  1-8. 

At  this  point  before  taking  up  the  further  detailed  treat- 
ment of  the  subject  matter  of  geometry,  it  may  be  well  to 
consider  the  form  of  recitation  on  theorems  and  originals, 
best  suited  to  the  new  conditions  under  which  geometry 
must  now  be  taught. 

3.  Form  of  a  Typical  Recitation  on  Theorems. — When 
the  study  of  the  propositions  demonstrated  (or  solved)  in 
the  text  is  begun,  it  is  of  course  important  to  make  clear  to 
a  pupil  the  meaning  and  parts  of  a  proof  and  also  to  teach 
him  how  to  study  a  proof.  He  is  to  be  brought  to  see  that 
mere  repeated  reading  of  a  theorem  and  its  proof  usually 
will  not  give  real  mastery  of  a  proposition,  but  that  he  is  to 
study  the  proof  till  he  understands  it,  then  try  to  write  it 
out  on  paper;  and  afterward  compare  his  written  work 
with  the  demonstration  as  given  in  the  textbook.  If  he 
finds  his  proof  is  faulty,  he  is  to  repeat  the  process  till  he  can 
prove  or  solve  the  proposition  correctly. 

It  is  well  sometimes  to  have  pupils  hand  in  at  the  beginning  of  the 
recitation  the  written  proof  which  is  the  result  of  their  study,  with  the 
following  or  a  similar  statement  attached:  "I  have  written  the  above 
proof  without  consulting  the  book  or  receiving  any  aid  from  any  person," 
or  more  briefly,  "no  help,"  or  "a  little  help."  Some  pupils  seem  to 
need  a  definite  end  like  this  to  work  for. 


4  TEACHING  OF  GEOMETRY 

We  shall  now  give  a  statement  of  what  we  consider  as, 
on  the  whole,  the  best  general  method  of  conducting  a  reci- 
tation on  text  propositions,  and  follow  this  statement  by  a 
mention  of  some  variations  from  the  general  method  which 
will  tend  to  stimulate  interest  and  add  further  effectiveness 
to  the  work. 

If  a  class  has  a  lesson  of,  say,  three  text  propositions  in 
advance  and  three  in  review,  good  results  are  obtained  by 
having  the  proof  of  each  advance  proposition  written  on 
the  blackboard  at  the  beginning  of  the  recitation  by  at 
least  two  members  of  the  class.  While  these  six  pupils 
are  engaged  in  thus  writing  the  proofs  on  the  board,  have  the 
figures  for  the  three  review  propositions  put  on  the  black- 
board where  all  the  rest  of  the  class  can  see  them  readily  (a 
teacher  can  often  save  time  by  himself  drawing  the  diagrams 
on  the  board  before  the  recitation  begins;  he  thus  also  has 
an  opportunity  to  stimulate  interest  by  varying  the  diagrams 
in  form,  position,  and  lettering).  Then  let  the  pupils  who 
are  at  their  seats  demonstrate  the  review  propositions 
orally  from  the  figures  thus  drawn.  It  is  often,  or  even 
usually,  best  to  divide  each  oral  recitation  of  a  proof  into 
two  parts,  each  part  assigned  to  a  different  pupil:  viz., 
1st.  A  careful  and  accurate  statement  of  the  general  and 
particular  enunciation  of  the  proposition;  2d.  The  proof 
proper  (or  in  case  of  a  problem,  the  construction  and 
proof). 

By  this  means  the  learning  of  the  exact  language  of  a  proposition, 
and  the  acquiring  of  a  clear  grasp  of  hypothesis  and  conclusion,  which 
are  matters  likely  to  be  neglected  and  slurred  over  by  the  pupil,  are 
brought  out  into  due  prominence.  (Hence,  for  instance,  it  is  not  neces- 
sary to  have  special  pads  or  forms  on  which  pupils  do  their  written 
work,  in  order  to  attain  this  end.)  Also  the  above  method  is  an  aid 
in  teaching  a  large  class,  since  more  pupils  are  called  on  to  recite  in  a 
given  time;  also,  a  weak  pupil  who  has  difficulty  in  making  demonstra- 
tions, soon  learns  to  enunciate  propositions  clearly  and  with  under- 


TEACHING  OF  GEOMETRY  5 

standing,  and  thus  lays  a  good  foundation  on  which  to  base  the  learning 
of  demonstrations  proper. 

By  the  time  that  the  oral  recitation  of  the  review  proposi- 
tions has  been  finished,  the  written  work  on  the  board  will 
be  ready  for  inspection. 

Time  may  often  be  saved  by  using  the  following  plan  in  inspecting 
such  written  work  on  the  blackboard.  Read  aloud  one  written  state- 
ment of  each  proposition,  carefully  correcting  or  having  the  class  correct 
each  defect  in  the  proof  read,  and  making  sure  that  the  whole  class 
understands  each  step  of  the  proof.  The  duplicate  proof  or  proofs 
may  then  be  read  silently,  or  quickly  glanced  over,  by  the  teacher, 
errors  being  pointed  out  and  corrected,  by  the  teacher  if  the  time  is 
short,  or  by  the  class  if  time  is  available. 

If  this  plan  of  conducting  a  recitation  on  text  theorems 
be  followed,  an  interval  of  from  ten  to  fifteen  minutes  will 
usually  be  left  at  the  end  of  the  recitation  which  can  be 
utilized  for  various  purposes,  after  the  next  lesson  has  been 
announced,  as  for  review,  for  sight  work  in  originals,  or  for 
written  work  on  paper  by  the  whole  class.  These  methods 
of  utilizing  the  last  ten  or  fifteen  minutes  of  the  recitation 
hour  will  be  described  in  more  detail  (see  §  5  of  this  pamphlet) 
after  we  have  given  some  methods  of  varying  the  first  part 
of  our  typical  recitation  on  text  propositions. 

4.  Variations  in  the  Methods  of  Conducting  a  Recita- 
tion on  Text  Propositions. — The  variations  next  to  be  men- 
tioned are  not  merely  useful  as  a  means  of  arousing  interest 
but  are  often  necessary  to  a  degree  owing  to  radical  differ- 
ences in  aptitudes  of  teachers  and  classes,  and  to  various 
other  circumstances. 

(1)  In  the  oral  recitation  of  review  propositions  it  is 
well  occasionally  to  dispense  with  diagrams  drawn  on  the 
blackboard.  Have  the  pupil,  when  reciting,  picture  and 
letter  his  diagram  mentally  and  have  the  class  follow  him 
in  his  proof  and  correct  every  misstatement;  or  after  a 


6  TEACHING  OF  GEOMETRY 

figure  has  been  drawn  on  the  blackboard  and  used  once, 
rub  it  out  and  have  the  pupil  give  the  proof,  using  mentally 
the  figure  as  it  had  beeri  drawn  and  lettered  on  the  black- 
board. 

(2)  Send  the  whole  class  (if  it  is  not  too  large)  to  the 
blackboard  and  have  each  member  of  the  class  write  on  the 
board  the  proof  of  one  theorem  after  another  during  the 
entire  recitation  period. 

(3)  Use  the  recitation  period  in  a  written  test  (on  paper) 
not  only  on  the  advance  work,  but  on  the  last  ten  (or  more) 
propositions  that  have  been  studied. 

(4)  If  the  class  is  large  have  part  of  the  class  do  written 
work  on  paper  at  their  seats,  while  the  other  part  of  the 
class  does  written  work  (on  originals  often)  at  the  black- 
board. 

5.  Methods  of  Using  the  Final  Ten  or  Fifteen  Minutes 
of  the  Typical  Recitation  on  Text  Theorems. — In  the  first 
few  lessons  in  the  study  of  theorems  and  originals,  a  consider- 
able part  of  the  final  fifteen  minutes  must  be  used  in  explain- 
ing the  nature  of  a  proof,  methods  of  study,  etc.  (See  §§3, 
6,  7.)  After  that  the  work  may  be  varied  in  one  of  the  fol- 
lowing ways: 

(1)  Have  the  entire  class  write  out  the  statement  and 
proof  of  one  or  more  propositions  or  originals  on  paper  at 
their  seats,  such  written  proof  to  be  carefully  corrected 
later  by  the  teacher  in  red  pencil  or  ink  and  returned  to  the 
pupil  at  the  next  recitation. 

It  will  be  a  considerable  economy  of  the  teacher's  time  and  strength 
to  indicate  the  proposition  to  be  proved  by  the  entire  class,  not  by 
writing  on  the  blackboard  the  proposition  to  be  proved,  but  by  drawing 
on  the  board  the  figure  (or  using  one  of  the  figures  already  drawn  there) 
belonging  to  this  proposition  and  having  the  class  write  out  the  theorem, 
enunciation,  and  proof,  all  members  of  the  class  using  the  figure  as 
lettered  on  the  blackboard.  The  economy  to  the  teacher  comes  from 


TEACHING  OF  GEOMETRY  7 

the  fact  that  in  correcting  the  proofs  the  teacher  quickly  commits  this 
lettering  to  memory  and  is  able  to  correct  a  paper  in  half  the  time 
otherwise  required. 

While  the  members  of  the  class  are  working  at  their 
seats,  an  opportunity  is  afforded  the  teacher  for  a  rapid 
grouping  and  inspection  of  the  papers  which  contain  the 
work  done  outside  the  class  and  which  have  been  handed  in 
at  the  beginning  of  the  recitation.  If  any  member  has 
failed  properly  to  do  this  work,  he  can  at  once  be  called  to 
the  desk  and  assigned  to  deficiency  study.  (See  §  13.) 

(2)  The  whole  class  may  be  sent  to  the  blackboard  to 
write   out   demonstrations.    This   procedure   often   has   a 
peculiarly  stimulating  effect  on  the  class.     It  seems  to  arouse 
in  a  striking  way  a  sense  of  unity  and  co-operation. 

(3)  While  the  class  remain  at  their  seats  the  teacher 
may  stand  at  the  blackboard  and  rapidly  sketch  one  after 
another  the  diagrams  used  in  the  last  fifteen  or  twenty 
propositions  and  ask  individual  pupils  to  state  the  proposi- 
tion relating  to  each  diagram. 

The  teacher  may  also  in  case  of  a  certain  diagram  ask  a  pupil  to 
draw  the  required  auxiliary  lines,  and  to  start  the  proof,  or  even  to  give 
an  outline  of  the  proof. 

(4)  The  time  in  whole  or  part  may  be  spent  in  simple 
construction  problem  exercises.     If  the  class  is  small,  all 
pupils  may  be  sent  to  the  blackboard.     If  the  class  is  too 
large  for  this,  some  of  the  pupils  may  work  at  the  black- 
board and  others  at  their  seats  doing  the  work  on  paper. 

The  class  may  be  asked  to  draw  accurately  the  figures  used  in  recent 
propositions  or  to  review  previous  construction  problems  by  drawing 
accurately  the  diagrams  used  in  them.  Or  examples  like  the  following 
may  be  devised  and  used: 

Ex.  1.  Construct  a  quadrilateral  ABCD  in  which  AB  =  1  in., 
£C  =  1J  in.,  CD  =  2  in.,  AD  =  2J  in.,  and  the  diagonal  AC  =  1J  in. 


8  TEACHING  OF  GEOMETRY 

Ex.  2.  Construct  a  triangle  whose  sides  are  2J  in.,  If  in.,  and  1  in. 
Then  construct  the  three  altitudes  of  the  triangle,  using  the  concurrence 
of  the  three  altitudes  as  a  test  of  the  accuracy  of  the  work. 

In  construction  work  at  the  blackboard,  have  the  pupils  construct 
arcs  by  use  of  a  string  with  a  piece  of  crayon  attached  to  one  end,  the 
crayon  being  sharpened  to  a  point  and  held  perpendicular  to  the  board. 
The  room  should  also  be  provided  with  a  number  of  rulers  two  or  three 
feet  long  divided  into  inches.  In  constructing  on  the  blackboard 
diagrams  like  those  called  for  in  Exs.  17  and  18,  p.  54,  inches  should 
be  changed  into  feet. 

(5)  The  time  may  be  employed  in  doing  the  exercises 
given  as  sight  work  at  the  foot  of  different  pages  in  the 
textbook. 

The  pupils  may  be  asked  to  solve  these  exercises  at  the  blackboard, 
or  the  teacher  may  stand  at  the  blackboard,  draw  the  figure  for  each 
exercise,  and  have  the  pupil  state  the  solution,  the  teacher  doing  on 
the  blackboard  such  mechanical  work  as  may  be  necessary. 

(6)  Frequently  the  time  may  be  profitably  spent  on 
sight  work  obtained  from  outside  sources,  on  originals,  or  on 
numerical  exercises,  the  teacher  standing  at  the  blackboard, 
drawing  the  diagrams,  and  bringing  out  and  recording  what 
is  given  concerning  the  diagram  and  what  is  to  be  proved, 
the  actual  demonstration  being  oral.     See  remarks  on  oral 
drill  in  originals,  §  8. 

(7)  The  teacher  may  spend  part  or  all  of  the  time  dis- 
cussing simple  practical  applications  of  geometry.     (See  §  9.) 

(8)  The  final  interval  which  we  are  considering  may   be 
spent  in  rapid  oral  review  of  definitions. 

(9)  Or  it  may  be  spent  in  a  combination  of  two  or  more 
of  the  preceding  methods. 

Utilization  of  the  preceding  methods  of  varying  the 
form  and  content  of  a  recitation  are  especially  valuable  in 
dealing  with  the  type  of  pupil  described  in  §  1.  A  certain 
amount  of  change  and  unexpectedness  gives  spice  to  the 


TEACHING  OF  GEOMETRY  9 

work  and  arouses  to  the  maximum  the  natural  growth  proc- 
esses in  the  child. 

6.  First  Theorems. — On  taking  up  the  study  of  theorems 
(text-book,  p.  29)  let  us  remember  that  the  child  has  no 
idea  of  what  a  demonstration  is,  or  of  its  utility.  The  fol- 
lowing has  been  found  a  satisfactory  way  of  approaching, 
Prop.  I,  p.  29.  Draw  two  intersecting  lines  on  the  black- 
board. Say  that  an  engineer  has  two  intersecting  lines  like 
these,  and  wants  to  know  what  is  the  least  number  of  the 
four  angles  which  he  must  measure  in  order  to  know  the  size 
of  all  the  angles.  The  child  has  already  had  some  experi- 
ence in  measuring  angles  and  is  quick  to  answer  "  one 
angle."  (Let  the  teacher  mark  on  the  diagram  one  angle 
as,  say,  47°  and  ask  the  pupil  to  give  the  size  of  the  other 
angles.)  This  opens  the  way  for  the  statement,  "  Let  us 
prove  that  in  all  cases,  no  matter  what  the  size  of  the  angles 
of  intersection,  the  property  which  you  have  used,  viz.: 
that  vertical  angles  are  equal,  is  true/'  Let  the  teacher 
draw  the  figure  on  the  blackboard  and  write  out  the  proof 
exactly  as  it  is  in  the  textbook,  explaining  it  step  by  step. 
Then  erase  this  and  ask  some  pupil  to  write  out  the  proof 
on  the  blackboard,  using  a  diagram  lettered  differently. 

After  this  proof  has  been  worked  out,  draw  three  lines 
intersecting  at  one  point,  and  ask  how  many  of  the  six 
angles  thus  formed  must  be  measured  in  order  to  determine 
the  rest  without  measuring  them,  and  so  on. 

The  first  lesson  on  theorems  comprises  Prop.  I  and  Exs.  1-7  which 
follow. 

Proposition  II  is  approached  in  a  similar  way.  An 
engineer  wishes  to  determine  whether  two  given  triangles 
are  equal.  He  measures  two  corresponding  sides  in  the 
triangles  and  finds  each  of  them  to  be  123  ft.  (Insert  each 
number  when  named  in  the  proper  place  in  a  diagram  drawn 


»  10  TEACHING  OF  GEOMETRY 

on  the  board.)  Then  two  other  sides  and  finds  each  of  them 
to  be  117  ft.;  then  the  angles  between  these  pairs  of  sides 
and  finds  each  of  them  to  be  63°.  Does  he  need  to  measure 
and  compare  any  more  parts  of  the  triangle?  The  instant 
reply  from  the  class  is,  "  No."  This  again  opens  the  way 
for  the  teacher  to  say  "  Let  us  prove  that  this  is  so." 

In  the  next  lesson  Proposition  III  is  treated  in  like 
manner. 

The  second  lesson  in  Book  I  may  comprise  the  definitions  on  pp.  30- 
32,  and  Prop.  II,  with  a  review  of  Prop.  I.  Exs.  1-10,  p.  32,  may  be 
treated  as  sight  work.  The  third  lesson  will  be  Prop.  Ill,  and  Exs.  1 
and  2,  top  of  p.  35,  and  a  review  of  Props.  I  and  II. 

In  introducing  Proposition  IV  (p.  37),  we  may  say  that 
an  engineer  knows  that  AB  and  BC  are  each  110  ft. ;  does  he 
need  to  measure  angles  A  and  C  in  order  to  determine  whether 
they  are  equal?  A  similar  method  may  be  used  with  advan- 
tage in  many  places  throughout  Plane  Geometry.  Thus,  in 
introducing  the  subject  of  parallel  lines  draw  two  parallel  lines 
intersected  by  a  transversal  and  ask  how  many  of  the  eight 
angles  formed  the  engineer  must  measure  in  order  to  know 
the  rest.  So  when  first  treating  similar  triangles,  draw 
pairs  of  triangles,  assign  numerical  values  to  certain  sides 
and  angles  and  ask  whether  each  pair  of  triangles  is  similar. 

7.  First  Originals. — After  studying  Propositions  I,  II, 
and  III,  it  is  well  next  to  stimulate  the  growth  processes  in 
pupils  and  to  arouse  the  pleasure  which  comes  from  a  sense 
of  mastery  by  having  the  class  prove  a  number  of  simple 
originals,  by  the  use  of  Propositions  II  and  III.  If  left  to 
himself,  the  pupil  will  usually  try  to  prove  these  exercises 
by  placing  one  of  two  given  triangles  upon  the  other.  This 
is  natural  since  it  is  the  only  method  of  proving  triangles 
equal  that  he  has  seen.  Hence,  before  asking  him  to  prove 
such  exercises,  it  is  well  to  have  a  preliminary  blackboard 


TEACHING  OF  GEOMETRY 


11 


drill,  in  which  the  teacher  draws  in  succession  many  pairs 
of  triangles  in  different  positions,  with  the  equal  parts 
written  on  the  triangles,  and  asks  whether  each  pair  is 
equal  and  why.  The  following  are  illustrations: 


H" 


14" 


(1) 


(2) 


In  some  cases  the  corresponding  parts  of  triangles  as 
given  should  not  be  equal,  and  the  pupil  should  be  asked 
to  change  the  given  numbers  so  as  to  make  the  triangles 
equal. 

The  first  originals  assigned  should  be  as  simple  and 
clear  as  possible.  The  work  can  hardly  be  made  too  easy 
at  the  start.  Do  not  hesitate  to  do  all  the  preliminary 
drudgery  for  the  pupil.  Draw  and  letter  the  diagram; 
state  the  parts  given,  and  to  be  proved,  leave  to  the  child 
the  pleasure  of  merely  discovering  the  proof.  Arouse  in 
the  child  the  pleasure  of  attaining  large  results  from  small 


12  TEACHING  OF  GEOMETRY 

effort.  As  his  powers  grow,  it  will  be  an  added  pleasure 
to  him  later  to  do  more  difficult  work,  as  by  converting 
abstract  language  into  a  diagram,  or  by  supplying  the 
definite  hypothesis  and  conclusion.  Step  by  step  he  will 
naturally  grow  into  the  power  of  dealing  with  triangles 
that  overlap,  of  proving  their  unknown  parts  equal,  and  of 
devising  auxiliary  lines  by  which  to  obtain  a  demonstration. 
At  the  end  of  a  month's  study  of  geometry  in  this  way, 
ninety  per  cent  of  a  class  proved  the  following  in  a  written 
test  with  the  greatest  zest  and  pride:  Given  A B  and  CD, 
two  lines  intersecting  at  0,  AC  parallel  to  DB  and  A0  =  CO; 
prove  OD  =  OB.  This  is  a  result  far  beyond  that  ever 
obtained  by  the  same  teacher  by  use  of  the  old  forcing 
process,  when  the  beginning  in  the  study  was  made  in  a 
much  more  difficult  and  formal  way. 

The  first  lesson  on  originals  may  comprise  Exs.  1-7  in  Group  9. 
In  the  next  lesson  Exs.  8-11  may  be  given  along  with  Prop.  IV. 

8.  Typical  Recitation  on  Originals. — In  order  to  obtain 
the  best  results  the  typical  recitation  as  described  in  §  3 
should  be  modified  when  teaching  originals.  At  the  begin- 
ning of  a  recitation  on  originals,  when  the  written  work  done 
by  pupils  during  the  last  fifteen  minutes  of  the  preceding 
recitation  is  returned  with  corrections  to  the  class,  there 
should  be  some  discussion  of  this  work.  When  this  dis- 
cussion is  concluded,  the  proof  of  the  originals  assigned  as 
the  advance  lesson  should  be  written  on  the  blackboard  by 
the  pupils  who  have  succeeded  in  solving  them.  While 
this  is  being  done,  the  part  of  the  class  at  their  seats  may  be 
employed  in  oral  review  of  originals  previously  studied,  or 
in  the  study  of  originals  which  are  to  form  part  of  the  next 
lesson.  After  the  work  written  on  the  blackboard  by  pupils 
has  been  discussed,  and  the  next  lesson  explained  and 
assigned,  during  the  last  fifteen  minutes  on,  say,  alternate 


TEACHING  OF  GEOMETRY  13 

days,  have  the  class  work  from  three  to  five  originals  at  the 
board.  The  largest  number  of  pupils  which  can  be  handled 
usually  in  this  way  is  about  twenty-five.  If  some  of  the 
pupils  finish  before  others,  sometimes  have  them  help  back- 
ward pupils  that  are  in  difficulty. 

On  other  days,  during  the  last  fifteen  minutes,  have 
pupils  do  written  work  at  their  seats.  When  a  pupil  in 
this  written  work  makes  some  glaring  fallacy,  or  devises 
some  specially  meritorious  proof,  it  is  well  to  call  attention 
to  these  facts  in  returning  the  corrected  papers  at  the  begin- 
ning of  the  next  recitation.  Thus  the  teachers  may  draw 
the  diagram  on  the  blackboard  and  ask  the  pupils  to  state 
bis  proof.  When  different  proofs  have  been  devised  for 
the  same  theorem,  it  is  peculiarly  instructive  and  stimulating 
to  the  class  to  have  all  of  these  proofs  presented.  See  for 
instance,  the  different  proofs  for  Ex.  4,  p.  82,  or  for  Ex.  7, 
p.  146. 

The  above  methods  of  teaching  originals,  with  modifi- 
cations which  will  suggest  themselves  to  the  thoughtful 
teacher  may  be  used  throughout  the  whole  subject  of  plane 
geometry. 

It  may  be  well,  however,  to  make  a  brief  statement  concerning 
special  drill  in  preparation  for  the  theorem  that  the  sum  of  the  angles 
of  an  n-gon  is  (2n— 4)  right  angles,  the  proof  of  which  is  difficult  for 
the  average  pupil.  In  the  next  lesson  after  proving  that  the  sum  of  the 
angles  of  a  triangle  is  two  right  angles,  draw  a  quadrilateral  on  the 
board  and  ask  for  the  sum  of  its  angles;  a  day  or  two  later  ask  for  the 
sum  of  the  angles  of  a  pentagon,  and  of  a  hexagon;  develop  the  fact  that 
an  n-gon  can  be  divided  into  (n-2)  triangles,  and  possibly  by  informal 
discussion  develop  the  formula  (2n-4)  right  angles  for  the  sum  of  the 
angles  of  an  n-gon.  Also  exercises  like  the  following  may  be  given. 
Draw  a  triangle,  denote  each  of  its  sides  by  a  and  one  of  its  angles 
by  x,  and  ask  the  pupil  to  determine  the  number  of  degrees  in  x. 
Also  draw  an  appropriate  pentagon  (equiangular,  not"  equilateral), 
denote  each  of  its  angles  by  x,  and  ask  the  pupil  to  determine  the  number 
of  degrees  in  x.  This  instance  shows  how  a  series  of  originals  may  be 


14  TEACHING  OF  GEOMETRY 

made  to  lead  up  to  an  important  theorem,  instead  of  having  the  reverse 
relation  of  following  the  theorem  as  applications  of  it. 

9.  Teaching  the  Practical  Applications  of  Geometry.— 
The  principle  of  giving  much  easy  work  at  first  and  utilizing 
natural  growth  processes  also  applies  when  instructing  a 
class  in  the  applications  of  geometry.  Feed  pupils  simple 
work  at  first,  and  they  will  naturally  grow  into  the  power  of 
doing  more  difficult  work.  It  is  worse  than  useless  to  pre- 
sent at  first  abstruse  or  technical  illustrations  of  the  utility 
of  geometry,  involving  scientific  or  engineering  principles 
quite  outside  the  scope  of  the  pupil's  past  experience. 
Even  an  example  like  Ex.  2,  p.  105,  if  used  too  early,  takes 
up  considerable  time  in  its  explanation  and  discourages 
and  repels  by  the  difficulty  and  unfamiliarity  in  its  applica- 
tion as  an  aid  in  measuring  the  velocity  of  light.  The  thing 
to  do  is  to  give  an  abundance  of  simple  applications,  till  the 
pupil  becomes  interested  and  begins  to  observe  and  suggest 
like  applications  for  himself,  and  finally  grows  into  an 
appreciation  of  more  difficult  cases.  Illustrations  of  the 
simple  applications  to  be  given  at  first  are  found  in  Ex.  10, 
p.  28;  Ex.  5,  p.  35;  Ex.  10,  p.  36;  Ex.  1,  p.  82;  Ex.  3, 
p.  106;  Ex.  20,  p.  86  and  Ex.  4,  p.  106;  Exs.  12  and  13, 
p.  132;  Exs.  1-4,  Group  38;  Ex.'  3,  p.  160;  Exs.  1-7,  p.  175; 
Ex.  16,  p.  177;  Ex.  7,  p.  186;  Ex.  9,  p.  197;  Exs.  1,  2,  3, 
p.  218;  Ex.  13,  p.  234,  etc. 

Of  particular  value  are  illustrations  which  the  teacher 
can  give  from  his  or  her  own  experience,  or  which  are  visible 
in  the  pupil's  own  life,  though  hitherto  unobserved.  The 
following  is  an  instance  which  I  have  often  used.  Formerly 
water  was  supplied  through  a  one-inch  pipe  to  the  house 
in  which  I  live.  Later  the  authorities  decided  to  replace 
this  by  a  two-inch  pipe.  When  I  saw  the  plumber  putting 
in  the  larger  pipe,  I  said  to  him,  "  That  is  good;  now  we 
shall  have  at  least  four  times  as  much  water."  "  No/7 


TEACHING  OF  GEOMETRY  15 

he  said,  "  you  will  have  twice  as  much."  I  took  a  sheet  of 
paper  and  drew  on  it  a  two-inch  circle  and,  inside  this  larger 
circle,  two  smaller  circles  each  one  inch  in  diameter  and  not 
overlapping,  thus  showing  that  the  area  of  the  larger  was 


more  than  twice  that  of  one  of  the  smaller  circles.  On 
seeing  this,  he  said,  "  I  guess  that  I  had  better  go  to  a  night 
school  and  learn  geometry." 

The  following  are  additional  simple  and  personally  observed  illus- 
trations of  the  practical  value  of  geometric  principles: 

In  a  house  heated  by  hot  air,  there  was  at  one  time  something 
wrong  with  the  furnace  and  pipes.  A  tinsmith  was  called  in  and  his 
first  move  was  to  determine  the  areas  of  the  cross-sections  of  the  various 
hot,  air  pipes,  in  order  to  compare  the  sum  of  the  areas  with  the  area  of 
the  cross-section  of  the  box  supplying  fresh  air  from  the  outside.  The 
ends  of  the  pipes  being  inaccessible,  with  a  tape  he  measured  the  cir- 
cumference of  each  pipe  and  from  the  length  of  the  circumference 
deduced  the  area  of  the  pipe.  This  shows  the  value  of  a  formula  for 
the  area  of  a  circle  in  terms  of  the  circumference,  or  better,  of  a  numer- 
ical table  from  which  an  area  may  be  obtained  when  a  circumference  is 
known. 

Or  again,  a  teacher  who  was  going  through  a  certain  village  happened 
to  see  a  boy  climbing  a  tree  and  carrying  a  stout  twine  up  with  him. 
On  asking  the  reason  for  this,  he  was  told  that  a  bet  had  been  made  as 
to  the  height  of  the  tree  and  the  boy  was  trying  to  measure  the  height 
by  carrying  a  twine  to  its  top  and  then  measuring  the  length  of  the 
twine  thus  used.  The  teacher  showed  him  that  this  labor  might  be 
saved  by  measuring  the  length  of  the  shadow  of  the  tree  and  also  the 
height  and  shadow  of  a  nearby  post. 

Or  take  this  problem  given  the  writer  by  a  sergeant  who  had  to 
solve  it  while  scouting  in  command  of  a  small  patrol  in  France  during 


16 


TEACHING  OF  GEOMETRY 


the  Great  War.  He  saw  a  column  of  the  enemy  moving  at  a  distance 
near  a  building  which  he  knew  to  be  120  ft.  high.  He  knew  that  the 
palm  of  his  hand  was  4  in.  wide  and  that  the  length  of  his  arm  from 
shoulder  to  the  bending  place  of  his  wrist  was  two  feet.  Using  these 
facts,  how  could  he  make  an  approximate  estimate  of  the  distance  of 
the  enemy? 

Similarly,  each  teacher  can  collect  a  series  of  personal  incidents 
illustrating  in  a  fundamental  way,  yet  one  that  takes  little  time  or 
effort  to  discuss  or  the  pupil  to  assimilate,  all  of  the  leading  princi- 
ples of  plane  geometry. 

Sometimes  it  is  advantageous  to  elaborate  at  some 
length  an  illustration  given  in  the  text.  Thus,  in  contin- 
uation of  Ex.  20,  p.  86,  draw  on  the  board  a  diagram  of  a 
pentagon  made  of  rods  hinged  at  the  vertices,  and  ask  how 
many  diagonal'  rods  must  be  inserted  in  order  to  make  the 
figures  rigid.  Ask  whether,  if  all  possible  diagonals  were 
inserted  and  fastened  together  at  all  possible  points,  the 
figure  would  be  made  still  more  rigid.  This  naturally 
raises  the  question  as  to  how  many  different  diagonals  a 
pentagon  has.  Treat  a  hexagon  in  like  manner.  Again, 


ask  whether  the  above  figure  A  BCD  made  of  jointed 
rods  (AB  and  DC  each  being  one  rod)  is  rigid;  and  if  not, 
how  many  diagonal  rods  must  be  inserted,  and  where  and 
why,  to  make  it  rigid. 


Also  ask  the  members  of  the  class  when  next  travelling  on  the  rail- 
road or  in  an  automobile,  to  observe  whether  the  trusses  of  all  bridges 


TEACHING  OF  GEOMETRY  17 

and  the  steel  framework  of  buildings  being  erected  are  not  made  up 
of  beams  and  rods  arranged  so  as  to  form  a  network  of  triangles.  If 
some  of  the  pupils  later  report  that  they  have  observed  frameworks 
which  were  in  part  composed  of  quadrilaterals,  this  opens  the  way  to 
ask  whether  the  engineer  has  shown  the  highest  type  of  skill,  and 
whether  the  framework 'would  not  be  stronger  if  cross  rods  had  been 
inserted  so  as  to  divide  all  quadrilaterals  into  triangles. 

10.  Ornamental  Designs. — Similarly  in  drawing  orna- 
mental designs,  such  as  the  tracery  of  stained-glass  windows 
and  architectural  outlines,  most  of  the  designs  usually  given 
at  first  are  too  complicated.  It  is  important  at  the  outset 
to  have  the  pupil  draw  many  simple  figures  till  he  becomes 
interested  and  develops  the  capacity  and  desire  to  attack 
the  more  elaborate  ones. 

In  this  connection  it  is  a  great  help  to  have  pupils  draw 
their  first  ornamental  or  architectural  figures  by  the  aid  of 
squared  paper.  In  doing  this,  pupils  are  stimulated  by  the 
fact  they  get  large  results  with  small  expenditure  of  time 
or  effort.  Also  by  noting  that  a  simple  array  of  parallel 
and  perpendicular  lines  can  be  so  useful,  they  get  a  new 
insight  into  the  utility  of  geometric  figures  in  general. 
This  kind  of  drawing  also  has  a  distinct  vocational  value, 
since  squared  paper  is  being  used  more  and  more  as  an  aid 
in  commercial  designing  of  artistic  forms. 

The  following  are  examples  of  the  simple  ornamental 
designs  which  may  be  used  at  first:  Ex.  11,  p.  36;  Ex.  6, 
p.  39;  Ex.  7,  p.  57;  Ex.  6,  p.  74;  Ex.  7,  p.  82;  Ex.  5,  p.  117; 
Exs.  6  and  9,  p.  120,  etc. 

When  desirable,  as  when  members  of  a  given  class  are 
especially  interested  in  ornamental  designing,  some  of  the 
above  examples  may  be  extended  so  as  to  form  other  similar 
designs.  Thus  after  a  class  has  constructed  a  trefoil  in 
the  manner  described  in  Ex.  6,  p.  120,  it  may  be  asked  to 
draw  a  square  and  on  it  construct  a  quatrefoil.  And  later 


18  TEACHING  OF  GEOMETRY 

to  make  regular  hexagons  and  pentagons  (see  Ex.  12,  p.  273) 
and  by  use  of  these  to  construct  sexfoils  and  cinquefoils. 

Also,  after  the  equilateral  gothic  arch  has  been  made  in  the  manner 
shown  in  Ex.  5,  p.  117,  the  pupil  may  be  taught  to  draw  lancet  Gothic 
arches  by  the  aid  of  isosceles  triangles  in  the  way  shown  in  the  diagram 


where  0  is  the  center  of  the  arc  AB.  If  the  pupil  is  required  to  prove 
that  the  tangent  to  the  arc  AB  at  B  is  perpendicular  to  the  base  BC, 
it  will  make  the  exact  form  of  the  figure  clearer. 

11.  Study  of  Numerical  Exercises. — Much  that  has  just 
been  said  concerning  the  teaching  of  original  exercises 
applies  also  to  the  teaching  of  numerical  exercises.  One 
or  two  additional  remarks  may,  however,  be  made.  The 
lack  of  power  on  the  part  of  the  average  pupil  to  see  a  numer- 
ical computation  as  a  whole  and  to  make  short  cuts,  to 
abbreviate  work  by  cancellation,  for  instance,  is  something 
extraordinary.  Some  of  the  principal  methods  of  shorten- 
ing work  are  mentioned  in  the  textbook,  pp.  282-283.  It 
requires,  however,  constant  watchfulness  and  insistence 
on  the  part  of  the  teacher  to  have  the  pupil  use  these  short 
cuts  till  they  become  natural  and  instinctive. 

Require,  then  that  the  pupil  draw  a  figure  for  every 
numerical  example,  that  he  denote  the  unknown  number 
whose  value  is  sought  by  x  or  an  appropriate  symbol.  (Do 
this  even  in  such  examples  as  Exs.  1,  2,  3,  etc.,  p.  284,  so  as 
to  fix  the  habit.)  Teach  him  to  group  or  indicate  together 
all  the  processes  involved  in  an  example;  to  use  cancellation 


.TEACHING  OF  GEOMETRY  19 

whenever  possible  (see  Exs.  19,  20,  etc.,  p.  285,  etc.);  and 
to  save  labor  by  suspending  operations. 


Thus  in  Ex.  5,  p.  287,  find  a  side  of  the  triangle  to  be  V^p  but  do 
not  reduce  this  result,  since  our  object  is  to  find  the  area,  and  the 
labor  of  root  extraction  at  this  stage  of  the  work  would  be  worse 
than  lost.  Also  in  Ex.  24,  p.  287,  let  x—  a  side  of  square  and  find 
a;2  =200,  and  use  this,  but  do  not  find  x  and  square  its  value  again  in 
order  to  find  x2. 

To  enable  some  classes  to  obtain  mastery  of  these  short- 
cut processes  it  may  be  advisable  for  the  teacher  to  take 
these  labor-saving  principles  one  by  one  and  to  dictate  a 
group  of  supplementary  exercises  for  each  principle,  every 
example  in  the  group  exemplifying  the  principle  in  hand. 

It  is  also  a  matter  of  some  importance  to  teach  pupils 
to  make  a  record  of  results,  if  such  results  are  likely  to  be 

of  use  later  on. 

\  . 

Thus  teach  them  to  record  on  a  fly  leaf  at  the  end  of  the  book,  and 
have  ready  for  future  use  such  results  as  the  numerical  values  of  V2, 
V3,  V5,  etc.;  or  of  x/2,  v^,  etc.,  in  Solid  Geometry.  This  leads  to 
an  important  saving  of  time,  and  at  the  same  time  instills  a  valuable 
educational  principle. 

In  all  the  earlier  work  with  numerical  examples,  the 
pupil  should  not  be  permitted  to  leave  his  final  results  in 
the  radical  form.  (Thus  for  Ex.  10,  p.  284,  the  answer  3 A/3 
should  not  be  accepted;  the  pupil  should  be  required  to 
carry  it  out  to  the  form  5.196+.)  After  the  pupil  has 
acquired  some  power  to  make  these  reductions  in  the  most 
advantageous  way,  time  may  be  saved  by  allowing  some 
results  to  remain  in  the  radical  form. 

12.  Algebra  an  Aid  in  the  Study  of  Geometry. — When 
branches  of  mathematics  are  studied  as  separate  subjects 
it  is  highly  valuable  to  have  a  cumulative  unification 
between  them.  Hence,  when  each  new  branch  of  mathe- 
matics is  taken  up,  it  is  important  to  review  and  unify  all 


20  TEACHING  OF  GEOMETRY 

preceding  branches  in  connection  with  it.  Thus,  when 
studying  algebra  it  is  desirable  to  review  arithmetic  as 
thoroughly  as  possible.  To  do  this  not  only  makes  the 
meaning  and  uses  of  algebra  clearer,  but  also  fixes  arith- 
metical processes  in  mind  and  gives  them  new  extensions 
and  useful  aspects. 

Similarly  on  taking  up  the  study  of  geometry,  the  two 
branches  of  mathematics  previously  studied:  viz.,  arith- 
metic and  algebra,  should  be  reviewed  and  applied  in  every 
practical  way. 

Thus,  one  of  the  parts  of  arithmetic  most  apt  to  be  forgotten  is  the 
topic  of  decimal  fractions.  Hence,  this  subject  should  be  made  promi- 
nent in  the  numerical  examples  given  in  the  applications  of  geometry. 

While  the  number  of  different  algebraic  processes  that 
can  be  advantageously  employed  as  aids  in  the  study  of 
geometry  is  not  large,  the  particular  forms  usable  are  notably 
valuable  in  that  they  give  room  and  call  for  initiative  and 
inventiveness  on  the  part  of  the  pupil  and  cultivate  the 
spirit  of  algebra  which  is  after  all  more  valuable  to  the  aver- 
age student  than  an  extended  knowledge  of  the  technique  of 
the  subject. 

Among  the  algebraic  processes  which  should  be  employed  where- 
ever  possible,  both  as  an  aid  in  the  rapid  and  thorough  mastery  of  geom- 
etry, and  as  a  training  in  algebra  itself,  are  the  following : 

(1)  The  use  of  a  single  letter  to  represent  an  angle,  line  segment, 
or  even  a  triangle.    As  examples -of  what  is  meant,  see  the  use  of  e,  /,  g, 
on  the  diagram  on  p.  29;  of  the  letter  a  on  p.  262;  of  7,  77,  777,  on 
p.  236.     It  is  curious  how  reluctant,  or  at  least  inactive,  pupils  are  in 
this  matter,  in  spite  of  its  manifest  advantages. 

(2)  The  use  of  marks  or  symbols  other  than  letters  in  indicating 
corresponding  parts  of  figures:     See  the  figures  on  the  next  page.     In 
the  same  connection  may  be  introduced  the  shading  of  over-lapping 
triangles  in  order  to  discriminate  them  clearly  from  the  rest  of  the  figure. 

If  the  matter  be  followed  up,  it  will  be  found  that  the  number  of 
ways  in  which  symbolisms,  formal  and  informal,  can  be  improvised  and 
used  to  advantage  is  remarkably  large. 


TEACHING  OF  GEOMETRY  21 

(3)  The  use  of  x,  or  of  some  other  letter,  to  represent  an  unknown 
magnitude,  as  in  Ex.  24,  p.  20;  Ex.  8,  p.  99,  etc.  (See  also  the  use  of  x 
in  solving  numerical  examples,  §11.) 


(4)  The  use  of  the  equation,  or  of  two  simultaneous  equations  as 
an  aid  in  solving  a  problem.     See  Ex.  24,  p.  20;  Ex.  8,  p.  99,  etc. 

Note  that  Ex.  24,  p.  20,  and  other  problems  may  be  solved  by  the 
use  of  simultaneous  equations. 

(5)  Transformations  of  the  formulas  of  geometry  to  obtain  new 
results. 

13.  Deficiency  Study. — Before  we  conclude,  it  may  be 
well  to  say  a  word  about  methods  of  bringing  up  the  work 
of  slow  or  negligent  students.  It  is  a  growing  custom 
among  schools  to  make  the  ordinary  recitation  period 
during  the  day  slightly  shorter  (to  reduce  it  say  from  45 
minutes  to  40  minutes)  and  thus  to  provide  time  for  an 
extra  period  at  the  end  of  the  day  in  which  special  attention 
may  be  given  to  laggards  of  any  kind.  This  we  may  call 
the  period  for  "  Deficiency  Study." 

There  are  many  different  ways  in  which  this  period  can  be  utilized 
in  teaching  geometry.  One  good  way  both  for  those  pupils  who  do 
not  know  how  to  study  and  for  those  who  have  neglected  their  work 
is  to  take  the  propositions  in  which  the  pupils  need  to  be  drilled  (say 
Book  I,  Props.  16-25),  divide  them  up  in  pairs  (Props.  16-17,  18-19, 
20-21,  etc.),  require  each  pupil  to  study  the  first  pair  till  he  thinks  he 
understands  them  (giving  such  help  as  may  be  wise),  and  then  require 


22  TEACHING  OF  GEOMETRY 

the  pupil  to  write  out  one  of  the  pair.  If  the  mistakes  made  are  serious, 
require  the  two  propositions  to  be  studied  and  one  of  them  written  out 
again.  So  continue  till  all  the  pairs  have  been  treated  in  like  manner. 
Drill  in  original  exercises  may  be  given  similarly. 

By  means  of  this  method  of  work  in  " deficiency  study"  quite  a 
large  number  of  pupils  may  receive  individual  attention  without  over- 
taxing the  teacher.  Competition  among  the  pupils  as  to  who  can  make 
most  rapid  progress  and  complete  the  work  soonest  will  also  be  a  material 
aid  in  the  work.  The  method  is  also  useful  to  the  teacher  in  making 
clear  what  pupils  have  been  dawdling  over  or  neglecting  the  work  of 
preparation.  If  a  pupil  in  deficiency  study  can  learn  the  demonstra- 
tions of  six  theorems  in  40  minutes,  and  comes  next  day  to  class  with 
only  one  proposition  prepared,  it  is  clear  that  the  pupil's  deficiency  is 
not  due  to  mental  dullness  but  to  lack  of  application,  and  the  teacher 
may  thus  know  how  to  proceed. 

14.  The  Study  of  Solid  Geometry. — The  remarks  which 
have  been  made  concerning  the  study  of  Plane  Geometry, 
apply  with  slight  modification  to  Solid  Geometry.  After 
the  teacher  has  tested  the  suggestions  made  with  respect 
to  the  former  subject,  most  of  the  modifications  necessary 
in  the  study  of  Solid  Geometry  will  occur  to  him  without 
special  mention  here. 

Much  time,  however,  may  be  saved  for  classroom  work  in  Solid 
Geometry  by  requiring  the  pupil  to  draw  on  paper  all  the  diagrams 
occurring  in  the  advance  lesson  and  to  hand  them  in  as  a  part  of  the 
preparation  of  the  lesson.  The  familiarity  with  the  diagrams  thus 
attained  not  only  facilitates  class  room  work,  but  is  also  an  important 
discipline  in  itself.  If  the  time  at  the  teacher's  disposal  is  small,  most 
of  the  original  exercises  on  construction  problems  may  be  omitted, 
and  formal  proofs  may  be  omitted  in  the  work  on  loci. 

The  least  amount  of  time  in  which  the  subject  of  Solid 
Geometry  can  be  mastered  with  any  degree  of  thoroughness 
is  a  half  year  with  five  recitations  a  week. 

In  the  pamphlet  entitled  "  Suggestions  Concerning  the 
Teaching  of  Algebra  with  Especial  Reference  to  the  Use  of 
Durell  and  Arnold's  Algebras/'  certain  other  details  will  be 
found  which  may  also  be  utilized  in  the  teaching  of  geometry. 


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